# How would you write mathematically that a random variable follows some unknown distribution?

For example, if I had a random variable X and knew it followed a normal distribution, I might write:

$$X \sim N(5, 10)$$

What would I write if I'm trying to say that the distribution is unknown?

$$X \in \Omega$$

Is that correct? It doesn't feel right.

• If you have previously defined or described $\Omega$ as a set of distributions that would be perfectly fine and clear. Because conventionally "$\Omega$" is already used to refer to a sample space, though, that might confuse an experienced statistical audience.
– whuber
Commented Feb 3, 2023 at 13:14
• I would just say that $X$ is a random variable. Commented Feb 3, 2023 at 21:25
• Unless there's some special reason you need to use mathematical notation, I would suggest to use narration instead: "Let $X$ be a random variable with [density $\delta$/cdf $F$/positive support/...]". Commented Feb 4, 2023 at 20:59

The notation I tend to see is something like $$X\sim F_X$$ to denote that $$X$$ is a random variable with $$F_X$$ as its CDF. I have seen people try to be brief and just write $$X\sim F$$, but this could mislead others into thinking that $$X$$ has an $$F$$ distribution, when that could be far from the case.

• Just to nitpick, we have books that go for $X\sim P_X$ too. Commented Feb 3, 2023 at 23:22

Standard notation is $$X\sim F$$ or $$X\sim F(x).$$.

Update: the latter notation, while common shorthand, could be misunderstood since $$F(x)$$ is a probability.

• I can't explain the downvote, but it made me look closely at this answer and I can't help noticing that "$X\sim F(x)$" is a mathematical solecism: $F$ might be a distribution function, but then $x$ must be a possible value of $X$ and therefore "$F(x)$" refers to a probability -- a single number.
– whuber
Commented Feb 3, 2023 at 19:46
• My, the nitpicking is strong with this community. Commented Feb 3, 2023 at 22:12
• @Stephan Nitpicking maybe--but that kind of abuse of notation is the source of plenty of misconceptions IMHO.
– whuber
Commented Feb 3, 2023 at 23:50
• @whuber that's why I started with F , and only as second option F(x). You could also write X~F(y). Hence why included it for clarity ( even though technically it's a probability). I doubt anyone who is familiar with probability/stats to the degree of sample spaces has a misconception (!) By F vs F(x) notation. I ll be more precise next time Commented Feb 4, 2023 at 3:04
• @whuber: I'm all with you on this, I have tried to get "nitpicker in chief" as a job title. I just found the five upvotes to your comment versus the zero net votes on the answer a trifle remarkable... Commented Feb 4, 2023 at 6:14